As President Barack Obama’s tax-cut plan runs into resistance in the House, Senate Democratic and Republican leaders say the upper chamber could move forward on a bill at any time, possibly even as early as Thursday.

“It’s my understanding it’s complete and ready and actually, we could move to that very soon,” Minority Leader Mitch McConnell (R-Ky.) said on the Senate floor.

Majority Leader Harry Reid (D-Nev.) agreed, saying, “We hope that we can very quickly lay down the tax bill.”

The leaders suggested the bill would be released this afternoon and the Senate could vote as early as Saturday to open debate, although a vote early next week looked more likely, according to a senior Democratic leadership aide.

There still appears to be work to do in the Senate on vote counting.

Reid sent a list of changes to McConnell Wednesday night that was aimed at winning over enough Democratic holdouts for a filibuster-proof majority. The proposed changes were described by sources as minor, but could pick up extra votes.

A Senate GOP aide said the leaders were trying to trim the size of the package and determine which tax extenders to include, although Republicans were insisting that the bill not become a vehicle to create new programs.

Senators and the White House have apparently agreed on a one-year extension of a key renewable energy grant program, according to key Democratic senators, a move that could attract undecided senators such as Maria Cantwell of Washington and Barbara Boxer of California.

A key ethanol tax credit has also made the cut as part of the last-minute changes, potentially drawing support from Sen. Tom Harkin (D-Iowa) and others.

White House Press Secretary Robert Gibbs said the White House estimates the bill would cost between $750 billion and $850 billion – about as much as the 2009 stimulus bill and the new health care law.

The quick movement in the Senate stood in contrast to the House, where the Democratic caucus voted Thursday to oppose the plan, throwing the deal into flux.

Oregon Rep. Peter DeFazio offered a resolution in a closed Democratic caucus meeting Thursday morning that said Democrats would oppose the bill in its current form. It is a nonbinding measure but illustrates Democratic displeasure with the bill that extends upper-income tax cuts and offers what they consider a generous deal to Republicans on the estate tax.

Gibbs said passing a bill through Congress is a “long and winding process,” but that the White House believes that the House will ultimately approve the bill extending the tax cuts before they expire at the end of the year.

“This is a game of calculus and physics, and I think the bottom line, though, is that we will have a vote that will not result in people's taxes going up by the end of the year,” he said. “I think at the end of the day, members are not going to want to be in their districts, senators are not going to want to be in their districts, when their constituents find out on the 1st of January that their taxes have gone up by several thousand dollars.”

“And I continue to believe that when all is said and done, if we don't get something done this year, everyone will I think rightly be blamed for not having gotten something done and we'll find ourselves, quite frankly, in a position where we're not getting the politics out of unemployment insurance for the rest of the year.”

Gibbs wouldn’t say if Obama would go to the Hill to press Democrats to pass the bill. “The president's been making his case, and the president will continue to,” Gibbs said.

As part of its continuing campaign to win passage of the package, top administration aides convened a conference call with reporters to lay out a new White House report outlining how the bill would help five hypothetical families.

White House Communications Director Dan Pfeiffer said the administration would “keep making the case to the House,” and that aides were in contact with the leadership in both chambers to address Democratic concerns.

Pfeiffer said the president had not received any promises from Republicans leaders on how many votes they could deliver, but that Obama was confident the bill would receive a “significant amount” of support from both parties.